An evolution compressible Stokes system in a polygon
The author investigates the evolution of a compressible Stokes system in a bounded polygon \(D\). The system describing the evolution has the form \[ u_t-\mu\Delta u+\nabla p=f \quad\text{in }D_\infty=D\times \mathbb R_+, \qquad \rho'(p_t+\beta\cdot\nabla p)+\rho\,\text{div}\,u=g \quad\text{in }D_\infty, \] under Dirichlet boundary conditions \(u=0\) on \(\partial D_\infty\), \(p=0\) on \(\partial D_{in,\infty}=\{x;x\in\partial D:\beta\cdot n<0\}=\partial D_{in}\times \mathbb R_+.\) The variables \(u\) and \(p\) represent the velocity and the pressure, \(\beta=[1,0]\) and \(f,g\) are given functions. The viscosity of the fluid is designated by \(\mu>0,\) \(\rho=\rho(\tau)\) means a smooth strictly increasing function of the ambient pressure \(\tau,\rho'=d\rho/d\tau,\) while the compressibility \(k=\rho'/\rho\) is a positive constant. The solution \(u(x,t)\) is considered as a mapping \(u(\cdot,t):\mathbb R_+\to X\), a Banach space, which is a function space on \(D\). Existence and uniqueness of the solution is proved, and several interesting estimates are obtained for the solution, in various norms. A splitting/decomposition for \(u\) and \(p\) is emphasized.
compressible Stokes system in a bounded polygon
scientific article; zbMATH DE number 2104515